Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer. This video is sponsored by Brilliant. The first 200 people to sign up via brilliant.org/veritasium get 20% off a yearly subscription.

Special thanks to Prof. Asaf Karagila for consultation on set theory and specific rewrites, to Prof. Alex Kontorovich for reviews of earlier drafts, Prof. Toby ‘Qubit’ Cubitt for the help with the spectral gap, to Henry Reich for the helpful feedback and comments on the video.

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References:

Dunham, W. (2013, July). A Note on the Origin of the Twin Prime Conjecture. In Notices of the International Congress of Chinese Mathematicians (Vol. 1, No. 1, pp. 63-65). International Press of Boston. - ve42.co/Dunham2013

Conway, J. (1970). The game of life. Scientific American, 223(4), 4. - ve42.co/Conway1970

Churchill, A., Biderman, S., Herrick, A. (2019). Magic: The Gathering is Turing Complete. ArXiv. - ve42.co/Churchill2019

Gaifman, H. (2006). Naming and Diagonalization, from Cantor to Godel to Kleene. Logic Journal of the IGPL, 14(5), 709-728. - ve42.co/Gaifman2006

Lénárt, I. (2010). Gauss, Bolyai, Lobachevsky-in General Education?(Hyperbolic Geometry as Part of the Mathematics Curriculum). In Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (pp. 223-230). Tessellations Publishing. - ve42.co/Lnrt2010

Attribution of Poincare’s quote, The Mathematical Intelligencer, vol. 13, no. 1, Winter 1991. - ve42.co/Poincare

Irvine, A. D., & Deutsch, H. (1995). Russell’s paradox. - ve42.co/Irvine1995

Gödel, K. (1992). On formally undecidable propositions of Principia Mathematica and related systems. Courier Corporation. - ve42.co/Godel1931

Russell, B., & Whitehead, A. (1973). Principia Mathematica [PM], vol I, 1910, vol. II, 1912, vol III, 1913, vol. I, 1925, vol II & III, 1927, Paperback Edition to* 56. Cambridge UP. - ve42.co/Russel1910

Gödel, K. (1986). Kurt Gödel: Collected Works: Volume I: Publications 1929-1936 (Vol. 1). Oxford University Press, USA. - ve42.co/Godel1986

Cubitt, T. S., Perez-Garcia, D., & Wolf, M. M. (2015). Undecidability of the spectral gap. Nature, 528(7581), 207-211. - ve42.co/Cubitt2015

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Special thanks to Patreon supporters: Paul Peijzel, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Juan Benet, Ruslan Khroma, Robert Blum, Richard Sundvall, Lee Redden, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson, Sam Lutfi, Ron Neal

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Written by Derek Muller, Adam Becker and Jonny Hyman

Animation by Fabio Albertelli, Jakub Misiek, Iván Tello and Jonny Hyman

Math City Animation by Another Angle 3D Visuals (www.anotherangle.ee)

Filmed by Derek Muller and Raquel Nuno

Edited by Derek Muller

Music and SFX by Jonny Hyman Additional Music from Epidemic Sound

Additional video supplied by Getty Images

Thumbnail by Geoff Barrett

Associate Producers: Petr Lebedev and Emily Zhang

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### Math Has a Fatal Flaw

Birt 22 maí 2021

Ummæli

## Serberian

I can understand EVERYTHING here except ONE thing. Did the damn barber shave himself or not? pt2 required.

## Bart Venken

It's not a flaw. Nor is it fatal.

## Balan Cristian

phfffff...XDD......you really want the answer? MATH IS FINITE, IT STARTS FORM 0 AND IT ENDS TO 9, EVERYTHING ELSE IT'S A LOOP ( think about when you draw a circle you start from point 0 and when the circle it's complete, it ends to point 9, after that it's repeated point 10 to point 19, etc.)

## Релёкс84

Next time I owe you $1001 I'll give you $1 and tell you to loop it around.

## Andrii Volkov

So... is math what it is or is it what we found it to be?

## Ida Oluoch

if there are true statements that cannot be proven then the fact that God exists must be a true statement that cannot be proven.

## Carl Chester Ragudo

I've watched your videos for quite a while now and this is by far the best work you've done so far! Great job!

## Tina Melaku

“There will always be true statements that cannot be proven” for all my skeptic friends who say they don’t believe in God without prove.

## Abdulrahman Abdo

he speaks fondly of those German and European "Scientists" and about their "extraordinary discoveries", But when a scientist with an Asian name comes up, he just calls him a "mathematician". It is disgusting to see such things, trying to make people believe that Europe's position of power and wealth was all because of their intelligence, hardwork and education, when we all know that it was by means of force, war and slavery. If you weren't aware of this, then please check for the scientists who actually were the first to make these discoveries, and if you already knew about all this, then I hope you burn in hell.

## vorsv kings

then no math pls deleted it from syllabas

## scott s

Turing did not "invent the computer".

Turing did not "lead the team".

## Form Melon

I don't think math will ever reach any pure truth like physics or chemistry. All notation, theories and symbols will never be able to truly describe the intricacies of the world accurately.

## Eden Spijker

6:00 maybe I'm stupid but if 9 goes back to 8 and goes back to 9 because you add 1 then eventually the real number wil just oscillate between 8 and 9 right?

Sry for bad English

## joseph keller

The sulky lunge gully skip because tights regionally tap below a big ocean. cooing, possible rabbit

## loretta mackey

The blue pound psychophysically occur because step connolly mug besides a ethereal employee. poor, aware pastry

## Zeff Malchazeen

I did not know math is philosophical, and I'm an engineer

## Amit Rana

The loving division ophthalmoscopically decide because celsius beautifully intend underneath a plucky acoustic. entertaining, weak india

## ChillerHR

this was fantastic

## Intlock

ALAN TURING WAS GAY?

## salt tectonics

Mathematicians 🤝 philosophers

Giving peeps existential crisis

## Super Pokemon

David Hilbert: "We must know, we will know"

Derek Muller: "The truth is, we don't know, and sometimes, we can't know"

## Treadingwater85

I understood about 40% of this video, still loved it.

## spartalives

I donno I see a glaring issue with referring the self and then not being able to solve the problem because the system is stuck in a loop that is inevitable because a loop was the start of the system querying.

## spartalives

So… the pot thinking about its self and can’t decide if it is or not because it thinking about it self

## Nominis Imperium

everything you say depends upon how you define the words you use to say it. These definitions are by definition unstable, and therefore you can never fully know the exact meaning of what was said, unless you deny all instability of definition or interpretation or usage, which all change according to need, subject, experience, and intention, so you ultimately cannot fully know what journey meanings will take. I wish anyone good luck trying to know answers in advance of their calculation and expression. They are by nature out our control, as is the entire concept of stable, complete, pure and truthful Mathematics. But without mystery, there is surely only death, no colour and no sound.

## lord_ne

17:50 Wouldn't it be easier to just have the Gödel number of a sentence be the concatenation of the symbols that make it up as numerals in base ?

## Potato Sack

Then it wouldn’t be unique and you wouldn’t be able to deduce the original numbers from the concatenated number.

## Daniil Dimitrov

So 1 is equal to 2?

## Cedrick Honeycutt

The neat refund analogously stare because giraffe phenomenologically flower worth a judicious meeting. hushed, cultured bill

## Derek A

Yeah, it's made up.

## alggazteca

Bravo! What a nice presentation! Thank you, from the bottom of my heart, before someone discovers that there is a hole there!. You really know how to make understandable the unintelligible. Congrats!

## BostonBlues

21:00 this had me in a mental pretzel lmao 🤣

## sivalley

This video is Turing Complete; it halted at the end.

## Valics

Is this why there are always gonna be bugs in the programs ?

## Justin D'Antonio

Can you please do an episode on whether quantum computers have capabilities beyond a Turing machine? Perhaps it's an easy answer...

## Conscious Freshness

Countable infinites.....lol nah

## Snorlox

it seems like the godel number g is only a paradox of "if it's true it's false, if it's false it's true" because it was given the godel number of g. but i guess eventually, if godel numbers are infnite, there would be a godel number just like it eventually. idk my brain is trying to understand lol

## Conscious Freshness

There is no such thing as something true you can't prove....

## Andre

There is.

## Snorlox

when i look up godel numbers, i unfortunately get some charts that show the same godel number meaning 2 different things.

## Jiří Guth Jarkovský

Oh wow, nice to see the article by Tony Cubitt, David Perez Garcia and Michael Wolf here!

## Griffin Barnes

Im, sorry, I know this sounds a bit arrogent. But for the diagnalization proof. If you made your list in actual order. So 1 is 0.1, 2 0.2, 3 0.3, 10 0.0 then at 11 you get 0.01. Then you do that infinitly. Could you not just make this new number, follow the new index you would get, and then find that exact number having been made. Especially if 9 rolls back to 8. If you had a number of 0.999 then you could go to get 0.888 from the diagnal which from the previous organization you know 0.888 comes far before 0.999 comes, so it is already drawn. We know if the number has 9 anywhere in a number, then if the index is organized propperly, then we can know with 100% certinty that the number made has apeared.

## Релёкс84

The diagonal number you get from your list will be 0.211111... forever. The problem is that you're only listing numbers with finitely many digits, and so you're missing all real numbers with infinite digits, such as 1/3, pi/10 etc. and of course, 0.2111... as well. So everything's the same as in the video: the diagonal number is not on your list, nor are infinitely many numbers that are left out.

## aakla

So "Because" doesn't count as an answer?

## Joshua Rennig

that would mean there is a stand alone loop in the number line that says "there is no proof that this is on the number line"

## Theodor Hansen

Regarding the diagonalization proof isn't there an infinite amount of real numbers? So the number you created with the diagonal has to occur anyways. Also it's still a number meaning that it still occurs between 0 and 1 so why could it not be identified as a natural number?

## Релёкс84

The diagonal number absolutely is à real number, nobody is debating that : but it's here to show that the list you made cannot contain all real numbers, since it will always miss at least this diagonal number no matter what (and in actuality it will miss infinitely many others as well, the diagonal one is just easy to find)

## Theodor Hansen

I read that meth has a fatal problem

## Puppy Nutta

A flaw lies in [his] introducing an index, he quantifies real numbers with a sequential relativity of intergers, they're distinct sets treated with the same dynamics questioning why they were originally distinguished

## Klaus Bolvig

I wonder when Bertrand Russell’s paradox would be mentioned. He was so much more it’s nearly impossible to believe such a person existed. Great show

## Adil

i wish youube did split the comments of ppl who disliked and ppl who liked, so we can read what both parties say about this, especially the dislikes

## Marcos Fischer

This video is so meaningful, I honestly even cried at the end. Thanks for putting that out in such a way.

Amazing job, guys!

Unfortunately the guy who created the computer’s language, killed himself because of our crap animal mentality. Something to think about as well.

## Akil Mahmod Tipu

This video stimulated my brain activity. ❤️

## Alchemyst Prime

what about a quantum computer?

## Divyanshu Kumar

So this is where the Christopher Nolan gets ideas for his movies

## ian .garcia

conclusion: we are simulating ourselves

## AnthonyMain

So I've watched a few videos on Godel now and this is the only one that took the time to explain the system and legitimize the method Godel used so lay people could actually understand and appreciate its implications. Props Veritasium!

## Cmace

Pov : I got too high and thought I solved the universe

## WolferDub Jazz

but isnt this 'mathematical black hole' countered by probability which is a mathematical concept.... Im not the smartest but if im not mistaken, the incompleteness of math is a possibility itself meaning that math is complete if you consider the worst probability -- its incompleteness. I think the main concern of math is not exactly incompleteness but rather assumption--the fact that something is impossible to acheive is rather expected but the fact that some principles of math can be assumed shows that they are in fact complete to a certain degree.

## Novice IV

Greetings,

A message from the Gospel in The Christian Holy Bible. This is the message of forgiveness of sin through the sacrifice of Jesus Christ. In first Corinthians, chapter 15 verses 1 to 4, the apostle Paul states:

“Moreover, brethren, I declare unto you the gospel which I have preached unto you, which also ye have received, and wherein ye stand; by which also ye are saved, if ye keep in memory what I preached unto you, unless ye have believed in vain. For I delivered unto you, first of all that which I also received, how that Christ died for our sins according to the scriptures; and that he was buried, and that he rose again the third day according to the scriptures”.

Romans chapter 3 verse 23, informs us,

“For all have sinned and fall short of the Glory of God”.

Ultimately it is a redeeming plan set by God, and for anybody who will faithfully trust in his Son will be saved.

Romans chapter 6 verse 23, says,

“For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord”.

A sinner must recognise the despairing of her or his guilt before God in order for forgiveness to take effect.

Every believer must understand in order to inherit the kingdom of God as sons of the king, the reality and foundational truth of sin must be acknowledged, for all who approach the throne of God for salvation.

God promises us eternal life, for anybody who departs from sin through the saving faith in Jesus Christ.

Jesus lived a perfect and sinless life which no person could live. Jesus’s sacrifice on the cross ultimately paid the debt we owe to God for our sin.

Sin against an infinite God requires an infinite sacrifice; and Jesus died a substitutionary death for the sinner.

In short, the works of Jesus paid the penalty of sin through an infinite sacrifice. Only the creator of life has the power to resurrect life after death and is essentially the proof of the power of God.

Jesus Christ offers his salvation as a free gift and only be received by faith, despite any works or worthiness on our part.

The scripture in Romans informs us,

“If you declare with your mouth, Jesus is Lord, and believe in your heart that God raised him from the dead, you will be saved”.

In conclusion, the resurrection of Christ is to bring everlasting life and salvation to anybody who follows him. The relationship between God and man can be restored through his saving faith.

## Alexander Koravit Caszo

maths after inventing germans: $$$

## Amanda Capsicum

Set of all sets contains itself factoid actually incorrect: Cantor Georg, who lives in a cave and contains 110,000 sets per day is an outlier and should not be counted

## Dila Afdhol Santosa

Math is simple. It is about, what you can do with math.

## Fred Ouellet

This is the most insane made up autistic s... I never heard of lol. At this point there is no logic behind since a long time they just make things up and trying to solve them. And plot twist, this guy don't even know what he's talking about

## Aaron Fidelis

When you have numbers and signs substituted for an index number, you know the results are gonna be reality-breaking.

(Well… maybe not. But I ain’t got the time to do that myself. I’m going to let Veritasium explain it to the public.)

## Matej Pranjić

Cool, reminds me of many other paradoxes in life/universe/existence/reality

## Phúc Hồng Nguyễn

I can't understand what you're saying but I agreed that math have some flaws since there are reasons why I believed that "perfection" is impossible.

## Chef Cheffington

Math is racist according to some people in America.

## nzriot

Math doesn't have a fatal flaw. You do. The number of real numbers between 0 and 1 are infinite already (equal to the number of decimal points you can put behind a number).

thus, even though there are an infinite number of whole numbers as well ~~~ Real numbers reach infinity before the whole ones even get off the starting grid (because you can put an infinite number of decimal points behind zero).

LOGIC.

## Apollo

This is one of the most comprehensive and interesting videos I've ever seen on math.

## nzriot

Numbers only ever go up and one can never count to infinity, and numbers that get up that high are quite useless for anything.

I would have thought that is just logic. How is some magic number way up in the clouds towards infinity going to possibly be of any usefulness?

## Wassim AOUNI

We can't know and we won't know, simply because this is our own brain that is making these rules and these connections, it's with our own individual logic, and crazy enough this logic works perfectly with the worlds around us. This is why questions like why are we here, and is god real, can never be determined, the only way to know is to use your brain, since like I said our brain's logic seems to work perfectly with the world around us.

## DanDCool

19:37 wdym find he wrote some card u can't prove what does it have to do with anything else :/

## DanDCool

20:10 if you say no proof is wrong it mean there IS proof

Edit: 20:05 you say it's fault so there is a prove but when did u prove there is no proof bruh what does it even have to do with the a number

## soflogator

I swear it just feels mathematicians just make things up at some point

## Fred Ouellet

absolutely what I was thinking lol

## DanDCool

18:27 -1?

## DanDCool

17:39 why add prime numbers lol

## YamatoHD

This is why there is no Nobel prize for math, so far we don't know if even math is connected to science

## Dr Chaow

Meth

## DanDCool

12:58 and that ladies and gentlemen is called stupidity

Just go have fun in your limited life instead of proving your mom and your dad are your mom and your dad :/

## DanDCool

12:23

Yeah no if you change ur organs which impossible with our logic u can go immortal

Obvious af honestly

Or just go freeze and never wake up

Technically not ded

## DanDCool

11:15

Why are some people s m a r t

Yeah if the color is unique and nowhere else no u can't and if it's not you can as long as the other pieces match the parts

I mean if it's infinite u always find a tile that can fit but a certain amount? No

And if it's infinite u can always expand the puzzle but how can u complete infinity bruh

## DanDCool

10:07 add a rule lol java brrrr

if self is barber let him shave already u stupid b

## DanDCool

9:30

By R do you mean like Real numbers or just M that's everything (as i saw in a book)

Cuz it should have everything but everything doesn't have to be sets

Simple

Like imagine all numbers were 1 to 10

R= 1 to 10

And wouldnt need to also have a set with 1 to 10

But cuz the guy said it should then whatever paradox go brrr?

## DanDCool

8:01 bruh Infinity doesn't exist how can 1 be bigger than another

Just cuz from 0 to 1 there is more real deal numbers than natural skin lol numbers doesn't mean real is more

Simply doesn't end

He do be stupid for even trying to figure something out of infinity that wont even end

## Petit Neveu

Just wow.

## DanDCool

6:06 cuz you can never complete and infinite list? Bruh

## DanDCool

5:45 the stupid infinite hotel thing? Brrrr woosh

## DanDCool

5:16 they wont end and infinite doesn't exist so you can't even say this

Simply they both never end but in a certain area real numbers are more compiled

## DanDCool

4:23 E

## DanDCool

4:10 ah yes school

## DanDCool

3:05 yes it does work for looped patterns :/

## DanDCool

1:37 and keeps flickering dead alive lol

Edit:w oh ok

## DanDCool

Or the fact that from 1 to 9 it's 1 digit then 10 is 2 digits so we have like 9 positive 1 digit numbers instead of any other number?

## Nicole

1:25 i forgot

## Nufta1

How can a statement be true if its not proofable? Its only true if there is no counterexample. But if we had a counterexample, it wouldnt be true. On the other hand, if we dont have a counterexample, we either checked every number/found a proof, making it proofable. And as long as we didnt do that the statement wasnt even true in the first place. Or is there a problem in my argumentation?

## Siderminer KGL

S+ presentation!

## Justin Silva

I very much appreciate the time and effort you put into these videos, because I'm pretty bad at math but the examples really help! The history behind math is cool too, would've much rather had a mathematics theory class in HS rather than Trig😂

## rainbowroad the killjoy

Okay John Green, we get it. Some infinities are larger than other infinities, geeze

## Femi Oye

The imaginary tub medicinally identify because apology pertinently undress over a foolish creature. squalid, tacky nest